Download SOLID STATE QUANTUM COMPUTING USING SPECTRAL HOLES

SOLID STATE QUANTUM COMPUTING USING SPECTRAL HOLES
M.S. Shahriar1, P.R. Hemmer2, S. Lloyd3, J.A. Bowers1, A.E. Craig4
1
Research Laboratory of Electronics, Massachusetts Institute of Technology, 77
Massachusetts Avenue, Cambridge, MA 02139
2
Air Force Research Laboratory, Sensors Directorate, Hanscom AFB, MA 01731
3
Department of Mechanical Engineering, Massachusetts Institute of Technology, 77
Massachusetts Avenue, Cambridge, MA 02139
4
The Spectrum Lab, Montana State University – Bozeman, Bozeman, MT 59717
A quantum computer that stores information on two-state systems called quantum
bits or qubits must be able to address and manipulate individual qubits, to effect
coherent interactions between pairs of qubits, and to read out the value of qubits.1,2
Current methods for addressing qubits are divided up into spatial methods, as when
a laser beam is focused on an individual qubit3,4,5 or spectral methods, as when a
nuclear spin in a molecule is addressed using NMR.6,7 The density of qubits
addressable spatially is limited by the wavelength of light, and the number of qubits
addressable spectrally is limited by spin linewidths. Here, we propose a method for
addressing qubits using a method that combines spatial and spectral selectivity. The
result is a design for quantum computation that provides the potential for a density
of quantum information storage and processing many orders of magnitude greater
than that afforded by ion traps or NMR. Specifically, this method uses an ensemble
of spectrally resolved atoms in a spectral holeburning solid. The quantum coupling
is provided by strong atom-cavity interaction. Using a thin disc of diamond
containing nitrogen-vacancy color centers as an example, we present an explicit
model for realizing up to 300 coupled qubits in a single spot. We show how about
100 operations can take place in parallel, yielding close to 4X104 operations before
decoherence.
The basic concept is illustrated in figure 1. Consider a small volume element of a
crystal containing a set of impurity atoms. Each atom sees a unique surrounding, so that
the resonance frequency for a given transition is different for different atoms. The number
of spectrally resolvable bands, NR, is determined by the ratio of the spectral spread to the
width of the individual resonance. We consider a situation where the number of atoms in
the selected volume is less than NR, so that each atom can be addressed individually.
We choose an effective density low enough to ignore the atom-atom direct
coupling. Instead, we couple the atoms in a controlled fashion by placing them in an
optical cavity with a strong vacuum Rabi frequency. Once two atoms are coupled by the
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cavity field, a variety of methods are potentially available for effecting quantum logic
between them: essentially any form of coupling between two spectral holes, combined
with the ability to perform single-hole quantum operations, allows the implementation of
general purpose quantum computations.8,9 The method we choose is determined by the
desire to perform two qubit operations accurately and with a minimum of decoherence.
This method is analogous to a scheme proposed by Pellizzari, et al.4 that uses adiabatic
transfer to move quantum information coherently from one particle to another, then
performs quantum logic by inducing single-particle Raman transitions.
Consider a situation where each atom has a Λ-type transition, with two nondegenerate spin states coupled to a single optically-excited state, as shown in figure 1. For
two atoms separated by a frequency matching the energy difference between the low lying
states, choose a cavity frequency that excites a resonance in each atom. Via this common
excitation, a cavity photon can act as a ‘quantum wire’ over which the atoms can
exchange optical coherence. Our qubits are stored on spins, however, and so we must use
optical coherence to transfer spin coherence. This is accomplished by applying, for each of
the two atoms, a laser beam coupling the remaining leg of the Λ transition. The resulting
two-photon excitation acts effectively as a cavity mode exciting the spin transition, with
the advantage that the excitation can be turned on or off at will by controlling the laser
beams. The atoms can use the two-photon mediated quantum wire to exchange spin
coherence with each other. If we tune the frequencies of the cavity as well as the laser
beams by the amount corresponding to the spin transition, we can couple one of these
atoms to a third one. In general, this scheme allows us to produce nearest-neighbor
information exchange among a discrete set of N atoms, where N is given by the ratio of
the inhomogeneous broadening to the spin transition frequency. Finally, M different spots,
spread in two dimensions, can be coupled by using the spatially selective version4 of this
technique, so that in principle up to MN qubits can be all coupled to one another.
In this scheme, each atom has a pair of Λ transitions and six low-lying spin states
(figure 2); for illustrative purposes it is convenient to think of these six states as resulting
from multiplexing of the spin states of two constituent pseudo-particles, a spin 1 particle
(A :purple) and a spin ½particle (B: turquoise) in each atom. In-between gate operations,
the logic states |0> and |1> of the qubit corresponding to every atom are stored in the spin
up and down states, respectively, of B, with A in the spin horizontal state, carrying no
information. Whenever it is necessary to perform a gate operation between two
neighboring gates, the information is first extracted from these storage levels. After this
restoration, the logic states |0> and |1> of the qubit corresponding to atom 1 are
represented by the spin up and down states, respectively, of A, with B in the spin up
state, carrying no information. On the other hand, the logic states |0> and |1> of the qubit
corresponding to atom 2 are represented by the spin up and down states, respectively, of
B, with A in the spin up state, carrying no information.
To entangle these two atoms, the quantum wire is used first to exchange, for
example, the quantum states of the A particles. This results in the atom 2 containing both
bits of information: 1 in B and 1 in A, unentangled. Quantum logic operations on the two
qubits now correspond to simple transitions between the spin sublevels inside atom 2.
Such transitions can be used to perform controlled-NOT gates and to entangle B and A.
The quantum wire is now used again to exchange the states of A, resulting in the
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corresponding entanglement of atom 1 and atom 2. By combining inter-atom quantum
wires and intra-atom quantum logic gates, it is possible to build up arbitrary quantum logic
circuits.
Under circumstances where the ground state splittings are very large compared to
the natural linewidth, this method works even when the two Λ-transitions (a-g-c and b-hd) are non-degenerate: εac ≠ εbd, but |εac−εbd| << εac . In this case, the cavity is detuned
away from resonance; the detuning must be large enough to ignore direct optical
excitation, but small compared to εac. The laser beam for each atom must have two
different frequencies, also detuned so that one of them is two-photon resonant with the ag-c transition, while the other is two-photon resonant with the b-h-d transition.
Figure 3 illustrates the steps used in producing an entangled state of the form
α|00> +β|11>, starting from the state α|00>+β|10>. More general entangled states can be
produced using these same steps. Figure 3a shows the process of retrieving the quantum
information from the storage levels. The curved arrows represent intra-atomic, Raman π
pulses, using laser beams only. Figure 3b shows the steps for producing intra-atomic
entanglement. First, a laser-cavity two-photon π pulse for atom 1 is used to transfer the
spin coherence from A1 (particle A in atom 1) to the quantum states of the cavity, as a
superposition of 0 and 1 photons. A second laser-cavity two-photon π pulse, now for
atom 2, transfers this information to A2. In practice, a counterintuitive pulse sequence
would be used to effect the same transfer adiabatically, which has the advantage of not
suffering from spontaneous emission during the transfer.4 An intra-atomic Raman π pulse
is now used to entangle A2 and B2. Finally, a reverse sequence is used to exchange A1 and
A2, producing inter-atomic entanglement. Figure 3c shows the final step of transferring
each qubit to its storage level, producing the desired state, which corresponds to a
controlled-NOT operation between the two qubits.
The technology that can be used to implement the proposed method is spectral
hole burning (SHB).10,11 In an SHB medium, the number of resolvable lines, NR, can be as
high as 107. Here, we present a specific SHB material for implementing this technique:
nitrogen-vacancy color center in diamond (NV-diamond).12 The relevant energy levels and
their correspondence to the model are illustrated in figure 4. As can be seen, both
degenerate (4a) and non-degenerate (4b) Λ transitions can be realized in this material.
While the degenerate case is conceptually simpler, in the case of diamond it has the
disadvantage that the hyperfine splittings are small, and comparable to the natural
linewidth, which in turns limits the maximum number of gate operations. As such, in what
follows we will concentrate on the non-degenerate case. While this system is not perfect, it
does meet the basic requirements, and allows us to discuss a concrete model of quantum
computing via spectral hole burning. A wide variety of SHB materials exist, including
quantum dots, and it may be possible to design media that are optimized for quantum
computing.
During the adiabatic transfer step, there are two primary sources of decoherence:
dephasing of the spin coherence and cavity losses. For example, the spin decoherence
time T2 in diamond is about 0.1 msec. Several NMR spectroscopic techniques exist for
increasing this to the T1 limit, and have recently been shown to be applicable to quantum
computing.13 While T1 has yet to be measured for NV-diamond, comparison with other
solids (e.g., Pr:YSO, where T2 is about 0.5 msec, while T1 is on the order of 100 sec) 14
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suggests that T1 is expected to be of the order of a few seconds. In addition to the NMR
type approach, one can use a diamond host free of the 13C isotope, which is known to be
the limiting source of spin dephasing.
Decoherence due to cavity losses can be minimized by using cavities with long
photon lifetimes. Specifically, we have estimated that a concentric, hour-glass cavity, with
a mirror separation of L cm, a waist size of D µm, and a Q of 2X105 can be used to
achieve, for diamond, a vacuum Rabi frequency of (170/DL1/2) MHz, and a cavity width of
(42/L) kHZ.15,16 Choosing D=50 and L=30, we get a Rabi frequency of 0.62 MHz, and a
cavity width of 1.4 kHz. The number of operations that can be performed before cavity
decay is more than 400. The cavity lifetime (1/2π*1.4 msec) is close to the T2, so that the
cavity still limits the maximum number of operations. The number of operations that can
be performed reliably could possibly be extended considerably beyond these numbers by
using error-correcting codes developed to circumvent cavity decay under similar
circumstances17,18.
The cavity design mentioned above may enable parallel operation, coupling many
different pairs simultaneously. The free spectral range (FSR) of the cavity would be ~250
MHz. By adjusting the value of L slightly, the FSR can be made to be a sub-multiple of
the channel spacing, which is about 2.8 GHz. Thus, in principle, all 300 channels could
operate simultaneously. However, in order to avoid undesired excitations, at least one
channel must be in the storage levels in-between two active pairs. This limits the
maximum number of parallel operations Np to about 100, so that the total number of
operations before decoherence can approach 4X104 even without error correction. Of
course, one must provide Np different control beams as well. In principle, this can be
achieved as follows: A series of acousto-optic modulators, each operating at 2.8 GHz, will
be used to generate Np sets of control beams from a reference set. The intensities of each
set can be controlled independently, and the beams can be combined using a holographic
multiplexer, for example19.
The controlled-not operation described above assumes exactly one atom per
spectral channel. To achieve this condition, one can start with a high dopant
concentration, in order to ensure that all channels have at least one atom. Consider first
the case of NV-diamond. For a 10 µm thick sample, the volume at the waist of the cavity
is about V=25000 µm3. For a dopant concentration of 1017 cm-3, the number of atoms is
four orders of magnitude bigger than the number of discrete channels. The probability for
having at least one atom per channel at an acceptable frequency would therefore be high.
The excess atoms in each channel can be removed by a gating process whereby a center
can be deformed via excitation at a frequency different from the transitions to be used, and
no longer responds to the optical excitation of interest20. Irradiation by a much shorter
wavelength (e.g., 488 nm) can restore the center to its desired form. Explicitly, the atoms
will first be pumped optically into one of the ground-state hyperfine levels. The cavity will
then be tuned to the corresponding leg of the Raman transition at a desired frequency. A
probe laser pulse tuned to the same transition will experience a shift in the cavity
transmission frequency21 proportional to the number of atoms in this channel. The
deforming laser, tuned to this channel, will be pulsed on, while monitoring the cavity
frequency-pull. The discrete step-size in the reduction of this frequency-pull will be used
to reduce the number of centers to unity. Since this deformation lasts for hours as long as
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the temperature remains below 4K, there will be virtually no time constraints in preparing
all the channels in this fashion. Finally, when considering atom-atom coupling, such as
dipole-dipole interaction, the parameter of interest is the mean distance, V1/3, between the
two atoms in adjacent channels. This distance is about 30 µm, much bigger than the laser
wavelength. As such, the atom-atom coupling can be neglected.
To extract information from the qubits, several techniques could be used. For
example, a high Q cavity could be used to detect whether an atom is in a particular ground
state sublevel by applying an optical π-pulse to an appropriate transition to drive it into an
excited state. Once excited, the atom can be probed by a variety of techniques such as
frequency pulling of the high Q cavity.21
In summary, we have proposed the use of spectral holeburning materials for
constructing quantum computers that have the potential to scale up to a large number of
qubits. For purposes of illustration, we have explicitly outlined the steps needed to
perform a quantum controlled-not using NV-diamond. For this system, upto 300 qubits
can be coupled, each to its nearest spectral neighbors, within a single spot. We also point
out how about 100 operations can take place simultaneously, yielding close to 4X104
operations before decoherence, even without error correction. This work was supported
by AFOSR and ARO.
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Figure Captions:
Figure 1: Schematic Illustration of coupling inhomogeneously broadened atoms using
spectral selectivity. The top figure shows a small volume of a crystal, selected by the
intersection of the cavity mode and the control laser beams. The bottom figure shows
how the atoms can be indexed in terms of their frequency response. Spectrally adjacent
atoms, with a frequency difference matching the ground state splitting, can be coupled
selectively by tuning the cavity and the coupling lasers. Atom M can be addressed
spectrally via the red transition, atom M+1 can be addressed via the green transition, and
the two are coupled to the cavity via the blue transition. The key constraint on the
matching is that the Λ transition in each atom must be two-photon resonant. This can be
realized by choosing the laser frequencies appropriately. It is also necessary to make sure
that there is only one atom per spectral channel.
Figure 2: Relevant energy levels and transitions required of two spectrally adjacent atoms
in this scheme. In each atom, the six low-lying levels can be thought of as corresponding
to the spin states of two pseudo-particles: a spin 1 particle (A: purple) and a spin ½particle
(B: turquoise ). In the quiescent state, the qubit in each atoms is represented by the spinup (0=e) and spin-down (1=f) states of B, with A in the spin-horizontal state, containing
no information. Whenever it is necessary to perform a gate operation between two
neighboring qubits, the qubit in atom 1 (α1|e>+β1|f>) is transferred to the spin-up and down states of A1, with B1 in the spin-up state (α1|a>+β1|c>). This pattern is alternated in
the subsequent atoms in the chain. The qubit in atom 2 (α2|e>+β2|f>) is transferred to the
spin-up and -down states of B2, with A2 in the spin-up state (α2|a>+β2|b>). Using a
sequence of pulses from the green and red lasers, the quantum states are exchanged
between A1 and A2, via the “quantum wire” provided by the blue cavity photon4.
Conceptually, this can be thought of as a two-step process. First, the red laser transfers
the state of A1 to the cavity, producing a superposition of 0 and 1 photons (α1|1>+β1|0>).
The green laser then transfers this state to A2 (α1|↓>+β1|↑>). All four bits of information
are now in atom 2; as such, any desired gate operation (see fig. 4) can be achieved by a
pulse coupling any two of the states (a,b,c,d), using a two-photon transition. A2 is now
exchanged with A1 by using a reverse sequence of the red and green lasers. Finally, the
state of each atom is transferred to the levels e and f. These storage levels are needed to
ensure that the neighboring qubits remain unaffected by these gate operations.
Figure 3: Illustration of the steps necessary to produce entanglement, starting from a joint
state (α|0>+β|1>)⊗|0>. (a) The state of each qubit is retrieved from the storage levels,
using off-resonance Raman π pulses, producing (α|a>+β|c>)⊗|a>. Either polarization
selection rules or an external magnetic field can be used to provide the selectivity of the
desired transition. (b) A pulse sequence of the red and green lasers exchanges, via the
common cavity photon, the states of A1 and A2 (see fig. 2), producing |c>⊗(α|c>+β|a>).
Another off-resonance Raman π pulse is used to transfer a to b, producing
|c>⊗(α|c>+β|b>); this is a controlled-NOT operation that entangles A2 and B2. A reverse
pulse sequence of red and green lasers exchanges back the states of A1 and A2, producing
(α(|aa>+β|cb>), which represents an entangled state of the two atoms. (c) The state of
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each qubit is now returned to the storage levels, producing the final state of
(α(|00>+β|11>), corresponding to a controlled-NOT operation between the two qubits.
Figure 4: Relevant subset of energy levels of the candidate material: N-V color centers in
diamond. The 3A1 to 3E transition is excited at 637.8 nm, with a homogeneous linewidth of
5 MHz and an inhomogeneous linewidth of 1 THz at liquid helium temperature. The
energy sublevels correspond to the spin orientations of the two uncoupled electrons (S)
and the nucleus (I) of the substitutional nitrogen atom. [A] The levels at zero magnetic
field. The inter-qubit frequency spacing in this case is 4.6 MHz, corresponding to more
than 105 qubits per spot. [B] The levels at a magnetic field of 500 Gauss, including nuclear
Zeeman splitting of 300 Hz/Gauss. The inter-qubit spacing in this case is 2.8 GHz,
corresponding to about 300 qubits per spot.
SINGLE
CAVITY
PHOTON
3
LASER 2
1
4
2
N
LASER 1
SINGLE
CAVITY
PHOTON
ooo
1
ooo
2
M
M+1
N
frequency
FIGURE 1
8
g
h
c
d
a
α1
e
α2
b
β1
g
h
c
d
a
b
e
β2
f
QUBIT 1
QUBIT 2
FIGURE 2
9
f
c
d
a
α
c
d
a
b
e
f
β
α
b
e
β
f
c
d
a
b
e
f
c
d
a
b
e
f
[A]
β
α
β
c
α
β
c
a
c
a
a
c
b
α
c
c
α
β
b
a
a
[B]
β
c
a
b
e
f
c
α
d
d
a
b
e
f
c
d
a
b
e
f
c
d
a
b
e
f
β
c
a
b
e
f
c
α
[C]
FIGURE 3
10
d
d
a
b
e
f
c
d
a
b
e
f
c
d
a
b
e
f
IZ
3E
IZ
1
g
h
∆f
∆f
SZ
-1
g
1
1
c
1
3A
1
d
a
e
0
1
b
f
-1
3A
1
0
c
d
4.9 MHz
b
4.9 MHz
2.8 GHz
-1
4.6 MHz
-1
∼5 MHz
h
3E
1
-1
a
1.4 GHz
0
e
2.8 MHz
SZ
1
[A]
-1
[B]
FIGURE 4
11
f
0.3 MHz
IZ
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